Paper Folding and Cutting — Study Notes

Overview

Paper Folding and Cutting is a visual reasoning topic in SSC CHSL that tests your spatial visualization ability. You are shown a sequence of paper being folded (once or multiple times), then cut with scissors or punched with holes. The question asks you to identify how the paper will look when completely unfolded.

This topic appears consistently in SSC CHSL Tier 1 with 1–2 questions per exam. While it seems abstract, mastering the underlying symmetry principles makes these problems quick to solve—typically under 30 seconds each. The key skill is mentally tracking how cuts replicate across fold lines. Unlike complex puzzles, paper folding follows strict geometric rules, so practice converts directly into marks.

Students who skip this topic lose easy points. The questions test pattern recognition and mirror-symmetry thinking rather than calculation. With 15–20 practice problems, most students achieve 90%+ accuracy. Focus on understanding fold axes and how each cut creates symmetric pairs or quadruplets of holes.

Key Concepts

• **Fold creates symmetry**: Every fold introduces a line of symmetry. A cut on the folded side creates mirror-image cuts when unfolded.

• **Count multiplier effect**: One fold = 2 layers (cuts appear twice). Two perpendicular folds = 4 layers (cuts appear four times). Three folds can mean 8 layers depending on fold type.

• **Fold sequence matters**: The order of folding determines the final pattern. A horizontal-then-vertical fold produces a different result than vertical-then-horizontal for the same cut position.

• **Cut position relative to folds**: A cut near the folded edge replicates close to fold lines. A cut at the corner of a twice-folded paper appears at all four corners when unfolded.

• **Shape preservation**: The shape of the cut (circle, triangle, square, semicircle) remains the same in all replicated positions. Only position and count change, not the cut shape.

• **Edge cuts vs. center cuts**: Cuts on the open edges (not folded edges) appear only on those specific edges. Cuts through multiple layers always replicate symmetrically.

• **Elimination strategy**: In MCQs, wrong options often show incorrect counts (too many/few holes) or break symmetry rules. Eliminate these first before detailed analysis.

Formulas / Key Facts

1. **Single fold (horizontal or vertical)**: 1 cut → 2 holes symmetrically placed across the fold line.

2. **Two perpendicular folds**: 1 cut → 4 holes arranged in a rectangular pattern (mirror in both axes).

3. **Two parallel folds**: 1 cut → depends on whether cut is between folds (3 holes possible) or through all layers (2 or 4 holes).

4. **Diagonal fold**: Creates 45° line symmetry. Cut replicates diagonally opposite when unfolded.

5. **Fold-in-half twice (same direction)**: Creates 4 layers in a strip. 1 cut → 4 holes in a line.

6. **Corner punch rule**: A hole punched at the closed corner (where all folds meet) appears at the center of the unfolded paper if folds are perpendicular and equal.

7. **Symmetry axes = number of folds**: One fold = one symmetry line. Two perpendicular folds = two symmetry lines (vertical and horizontal).

8. **Open edge cut**: A cut on the edge that is not folded appears only once in that location—no replication.

Worked Examples

**Example 1: Single Horizontal Fold**

*Problem*: A square paper is folded in half horizontally (bottom to top). A circular hole is punched in the center of the folded paper. How does it look unfolded?

*Solution*:

• Step 1: Folding horizontally creates 2 layers (top half over bottom half).
• Step 2: Punch creates a hole through both layers.
• Step 3: Unfold mentally—the hole in the top layer stays in the upper half; the hole in the bottom layer appears in the lower half at the mirror position.
• Step 4: Result = 2 circular holes placed symmetrically above and below the horizontal center line.

**Example 2: Two Perpendicular Folds**

*Problem*: A rectangular paper is first folded vertically (left to right), then folded horizontally (bottom to top). A small square is cut from the corner where all folds meet. What pattern appears when fully unfolded?

*Solution*:

• Step 1: First fold (vertical) creates 2 layers.
• Step 2: Second fold (horizontal) creates 4 layers (each of the 2 layers folded again).
• Step 3: Cutting the closed corner (where both fold lines meet) cuts through all 4 layers.
• Step 4: Unfold horizontally first—2 squares appear vertically aligned.
• Step 5: Unfold vertically—each of those 2 squares duplicates, giving 4 squares total arranged in a 2×2 grid pattern.
• Answer: 4 square holes at four corners of a smaller rectangle in the center.

**Example 3: Diagonal Fold with Edge Cut**

*Problem*: A square is folded diagonally (corner to opposite corner). A semicircular cut is made along the folded edge (the diagonal). How does the unfolded paper look?

*Solution*:

• Step 1: Diagonal fold creates symmetry along the 45° line.
• Step 2: Cutting along the folded edge means cutting through both triangular layers at the diagonal.
• Step 3: Unfold—the semicircular cut on the diagonal edge appears on both sides of the diagonal.
• Step 4: Result = A circular hole in the center (two semicircles join) OR two semicircular notches along the diagonal, depending on cut depth.
• Check answer choices for the symmetric diagonal pattern.

Common Mistakes

1. **Forgetting to count all layers → Undercounting holes**: Students often forget that each fold doubles the layers. Fix: Before solving, write down the total number of layers (1 fold = 2, 2 folds = 4, etc.) and ensure your answer has that many cuts.

2. **Mixing up fold order → Wrong symmetry placement**: Folding horizontally then vertically is NOT the same as vertically then horizontally for off-center cuts. Fix: Mentally unfold in reverse order—last fold opens first. Track each step separately.

3. **Placing cuts on the wrong side of the fold line → Mirror image errors**: Students often place replicated holes on the same side as the original cut instead of mirroring across the fold. Fix: Draw a mental (or actual) fold line and consciously place each hole on the opposite side at equal distance.

4. **Ignoring open edges → Adding extra holes**: Cuts on unfolded edges (free edges) do not replicate. Fix: Identify which edges are folded vs. open before predicting replication.

5. **Rotating shapes incorrectly → Orientation mistakes**: When a triangular or rectangular cut is made, its orientation must mirror correctly. Fix: Preserve the cut shape's orientation relative to the fold axis when mirroring.

Quick Reference

• **One fold = 2 holes** (mirror across fold line).
• **Two perpendicular folds = 4 holes** (rectangular symmetry pattern).
• **Count layers before predicting cuts**—number of layers = number of replicated cuts.
• **Unfold in reverse order**—last fold opens first when mentally visualizing.
• **Closed corner cuts** appear at the center or in symmetric grid positions.
• **Free edge cuts don't replicate**—only folded layers create multiple holes.